Economic growth and economic development 264

Introduction to Modern Economic Growth
5.12. Exercises
Exercise 5.1. Recall that a solution {x (t)}Tt=0 to a dynamic optimization problem

is time-consistent if the following is true: whenever {x (t)}Tt=0 is an optimal solution

starting at time t = 0, {x (t)}Tt=t0 is an optimal solution to the continuation dynamic
optimization problem starting from time t = t0 ∈ [0, T ].
(1) Consider the following optimization problem

max

{x(t)}T
t=0

T
X

β t u (x (t))

t=0

subject to

x (t) ∈ [0, x¯]
G (x (0) , ..., x (T )) ≤ 0.
Although you do not need to, you may assume that G is continuous and
convex, and u is continuous and concave.
Prove that any solution {x∗ (t)}Tt=0 to this problem is time consistent.

(2) Now Consider the optimization problem

max u (x (0)) + δ

{x(t)}T
t=0

subject to

T
X

β t u (x (t))

t=1

x (t) ∈ [0, x¯]
G (x (0) , ..., x (T )) ≤ 0.
Suppose that the objective function at time t = 1 becomes u (x (1)) +
P
δ Tt=2 β t−1 u (x (t)).
Interpret this objective function (sometimes referred to as “hyperbolic

discounting”).
(3) Let {x∗ (t)}Tt=0 be a solution to this maximization problem. Assume that

the individual chooses x∗ (0) at t = 0, and then is allowed to reoptimize at
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